Proximal flows of Lie groups (Q796679)

Let (G,X) be a G-flow, i.e. X is a compact Hausdorff and G is a locally compact group acting on X. This action induces an action of G on the space M(X) of probability measures on X endowed with the weak * topology, making (G,M(X)) a G-flow. A set \(Y\subset X\) is called minimal if \(\overline{Gy}=Y\) for every \(y\in Y\). A flow (G,X) is called proximal if \(\{(x,x);\quad x\in X\}\) contains all the minimal subsets of the flow \((G,X\times X),\) where \(g(x,y)=(gx,gy).\) It is called strongly proximal if the collection \(\{\delta_ x;\quad x\in X\}\) of point measures contains every minimal subset of \((G,M(X)).\) The author proves the existence of a flow (G,X) where G is a Lie group such that (G,X) is proximal but not strongly proximal. The proof is based on a result contained in the preprint of the paper by \textit{M. Ratner} [Ergodic theory in hyperbolic space].